Решения для пересечения трех бран и суперсимметрии Текст научной статьи по специальности «Математика»

UDC 530.1; 539.1

Triple M-brane solutions and supersyrnmetry

A. A. Golubtsova1 bc and V. D. Ivashchuk2 a b

(a) Center for Gravitation and Fundamental Metrology, VNHMS, 46 Ozyornaya Str., Moscow 119361, Russia (b) institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia, 6 Miklukho-Maklaya Str., Moscow

117198, Russia

(c) Laboratoire de Univers et Théories (LUTh), Observatoire de Paris,

Place Jules Janssen 5, 92190 Meudon, France

E-mail: (1) [email protected], (2) [email protected]

We study composite M-brane solutions in 11-dimensional supergravity. The supersymmetric solutions describing orthogonally intersecting M-branes are defined on the product of Ricci-flat manifolds Mi. The amount of preserved supersymmetries depends upon certain numbers of (chiral) parallel spinors on factor spaces Mi and brane sign factors. Three examples of triple M-brane configurations are considered and the numbers of unbroken SUSY are obtained.

Keywords: M-branes, parallel spinors, supergravitv, supersymmetrv.

1 Introduction

2 Generalized Killing spinor equations

Solutions to supergravity theories preserving some amount of supersymmeries play an important role in studies in non-pertubative M-theory and correspondences between gravity and gauge theory.

We consider intersecting M-brane solutions defined on the manifold of the form

M = M0 x M1 x ... x Mn

(1)

where Mi are Ricci-flat manifolds. In what follows we denote di = dimMi. In [1] the classification of super-symmetric M-brane configurations on product of flat factor spaces Rdi was presented and the relation for the amount of preserved supersymmetry was found:

N =2 k, with k = 1, 2, 3, 4, 5.

(2)

The bosonic action in 11-dimensional supergravity is given by

S = J d11z^m{R[g] - ¿(4)F2} - A A F A F,

(3)

where F = dA = 4yFNPQRdzN A dzP A dzQ A dzR is 4-form. We consider pure bosonic configurations in D = 11 supergravity (with zero fermionic fields) that are solutions to the equations of motion corresponding to the action (3). The amount of preserved supersymmetries (SUSY) corresponding to the bosonic background (eM, AMl m2m3) is defined by the dimension of the space of solutions to (a set of) linear firstorder differential equations (generalized Killing spinor equations) for 32-component spinor field e = e(z)

It was shown in [2] that for the simplest M5-brane configuration to obey the supergravity equations of motions, the brane world volume should be Ricci-flat and admit parallel spinors for supersymmetry. The relation (2) is no more valid if composite M-brane configurations on the product of Ricci-flat manifolds (1) are taken into consideration. In this case N depends upon certain numbers of chiral parallel (i.e. covariantly constant) spinors on Mi and brane sign factors cs = ±1.

In this note we study supersymmetric solutions defined on the manifold (1) for triple M-brane configurations in D = 11 supergravity. The case of two in-M

M2- and M5-brane solutions) defined on the product of Ricci-flat manifolds was considered earlier in [4]. (For solutions with one brane see also [3].)

+ Bm )e = 0.

Here

Dm = 9m + - w ABM r AfB

(4)

(5)

is covariant spinorial derivative, the gamma matrices satisfy the Clifford algebra relation

r ArB + rB rA = 2nABl 32, (6)

BM is an operator induced by the 4-form field strength F

BM = 288 rNrPrQrR - 12^NfPrRpR) Fnpqr

(7)

and rM are worId T-matrices.

The number of preserved SUSY is N = N/32, where N is the dimension of the linear space of solutions to differential equations (5).

3 Intersecting M-brane solution

The metric of intersecting M-brane solutions of D = 11 supergravity can be written in the following form

where

D m n = 0,

l = 0,... ,n and

r[s]V = csn, f°r all s G S.

(15)

(16)

+^2 e2^(æv, i=1

(8)

e27

( n Hs)1/3( n Hs)2/3,

e2^ = e2Y

n H-

seSe

seSn

seS

(9)

i = 1,. .. ,n.

Here = dmi + 4f “l fis the modified oper-

ator of covariant spinorial derivative corresponding to factor-space Ml with the spin-connection u(l).

4 Triple M-brane backgrounds

In this section we present three examples of solutions.

4.1 M5 n M5 n M5

The solution describing three intersecting magnetic M5

Here g0 = (x)dX ® dxv is a Ricci-flat metric on

the manifold M0 mid gl = gim^i(Vi)dy"H <8> dyn is a Ricci-flat metric on Mi, 1,.. .,n, 5iis = Sij is the

jeis

indicator of ¿belonging to Is: 5iIs = 1 for i G Is and SiIs = 0 otherwise.

The 4-form field strength reads

F = ^ CsdH-1 At(Is)+ ^ cs(*odHs) At(I,), (10)

Mo x M1 x M2 x M3 x M4,

where d0 = 1, d1 = d3 = d4 = 2, d2 = 4. The solution reads

(17)

seSe

SeSm

where c2 = 1, *0 is the Hodge operator on (M0,g0), Hs is a harmonic function on (M0, g0) and Is = {1,2,..., n} \ Is is a dual set.

The set of indices Se describes electric branes and Sm describes magnetic branes.

We put

r[s] =rAlrA2rA3, for s G Se, (11)

where (rAio )2 = —132 for some i0 G {1, 2, 3} (f Ai) for

i = i0

g = FpH2^3! ff0 + H-1^ +

H-1H-1H-1 ff2 + HfV + H-g4}, (18)

F = c1(*odH1) A T1 A T4 +

C2(*odH2) A T3 A T4 + C3(*odH3) A A T3, (19)

where c1 = c2 = c3 = 1; H1; H2, H3 are harmonic

functions on (M0,g0). The metrics gl (i = 0,1, 3,4)

g2

signature (—, +, +, +). The branes sets are I1 = {2,3}, I2 = {1, 2} I3 = {2,4}.

Using the rules of decomposition from [5] one can write T-matrices in the following form

(r A) =

[s]

rBlrB2rB3rB4rB5, for s G Sm,

(12)

where (rB )2 = 132 for all i.

It follows from the relations (11)-(12)

/ 1 r(1) r (2) r (3) ^(4) \

1 ?F ai (1) r (2) r (3) ^(4)

1 I2 r a 2 1 (2) r (3) ^(4)

1 I2 14 if a3 (3) ^(4)

V 1 I2 14 2 Pa4 1 (4) /

(r[s])2 = 132. (13)

In the recent work [4] it was shown that the relation (4) is satisfied identically if the spinor field e can be represented in the form

Here

r (1) =

obey

ff1l ff21 r (1)r (l),

r(3)

1 (2) =

p13 ff23 І(3)І(3),

ff 12 ff22 ff32 ff42

(2)^ (2)r (2)r (2),

ff . v _____ ff 14 ff 24

1 (4) = 1 (4)r (4)

-1/6

(r (2))2 = —14, (r(i))2 = —12

(20)

(21)

n Hs

n Hs

-1/12

(14)

lseSe

VseS„

with i = 1, 3, 4. r^1),rr^ are 2 x 2 T-matrices

M1 M3 M4

s

I

e

n

The covariant derivatives can be represented as

D m? = dmi + 1 w(1b m f 1 ® rai1)rb1) ® 14 ® 12 ® 1 2

mi ' 4 aibimi

1

Here nu i = 1, 2,3,4 are prnallel spinors defined on Mj. respectively ^D^in = 0^, obeying (23) and (24).

Eqs. (24) have the following solutions

Dm2 = dm2 + 4w{Xm2 ^ ® 12 ® rai)rb2) ® !2 ^2

Dm3) = dm3 + 4^^3 f1 ® !2 ^4 ® ra3)rb3) ® 12

-^2 ffb2

c(1)

or

ic2, c(2)

c(3)

ic^ c(4) — —ic3,

Dmi = dm4 + 4wati>4m4 (1 ® 12 ® 14 ® 12 ® ra44)r(4)) , c(1) = ic2, c(2) = —i, c(3) = ic1, c(4) = ic3.

(i)

w(a)b.c. is a spin connection corresponding to the man- The number of linear independent solutions given by

Mi i

1,2, 3,4, D^ml are

■ , , ■ (25), and reads

covariant derivatives '

corresponding to Mi, i = 1, 2, 3 4„ = dm0 and n = 32N = n1(—ic2)n2(i)n3(—ic1)n4(—ic3) +

D

(0)

3„

n1(ic2 )n2(—i)n3(ic1)n4(ic3), (26)

Let n = n0 ® n1 (y1 ) ® n2 (y2) <8> n3 (y3) <8> n4 (y4), where where nj (cj ) is the number of chiral parallel spinors on n0 is a 1-component spinor on M0, n2 = n2(y2) is a 4- Mj j = 1, 2, 3,4. component spinor on M2, ni = ni(yi) ^ a 2-component Mi i = 1, 3, 4 The following relations for modified covariant derivatives take place:

D(0)

4.2 M2 n M2 n M5

Let us consider the intersection of two electric and M2nM2nM5

on the manifold

Dmin = n0 ® (Dm!rn) ® n2 ® n3 ® n4, Dj^n = n0 ® n1 ® (Dm!ns) ® n3 ® n4,

M0 x M1 x M2 x M3 x M4 x M5 x M6,

(27)

4 n4

(22)

where d0 = 3 di = d2 = d3 = d4 = d5 = ^d d6 = 3. The metric and the 4-form field strength of two inter-M2 M5

sented in the following form

The operators (12) corresponding to

M 5-brmies g = H11/3h21/3h3/3{ g0 + Hi 1g1 + H2 1ff2 +

read

r[s] = r1or1i r2ir14r24 = 1 ® 12 ® r(2) <g> r(3) ® 12,

for S = I15

r[s] = r1or13r23r14r24 = 1 ® r(1) ® r(2) ® 12 ® 12,

for s = I2 and

r[s] = r1or1i r2i r13r23 = 1 ® 12 ® r(2) ® 12 ® r(4),

H-1H-1H-1ff3 + H-1H-1ff4 +

H2-1H3-1ff5 + H3-1g^.

The corresponding field strength is

F = c^H- A T1 A A T4 + c2dH-1 A T"2 A A T5 + c3(*0dH3) A A T2,

(28)

(29)

c21 = c22

1 H1 H2 H3

monic functions on (M0,g°). The metrics gj (i

s = I3

^ ^ ...„x 0,1,2,4, 5, 6) have Euclidean signatures and we put the

3

metric g3 = — dt®dt. lhe branes sets are /1 = {1, 3,4}, I2 = {2, 3, 5} I3 = {3, 4, 5, 6}

...........owing form

c2

c(j)

1,

r(j)nj = c(j)nj, c(.

j = 1, 2,3,4 and

c(2)c(3) = c1, c(1)c(2) = c2, c(2)c(4) = c3.

may be chosen in the fol

\A

(24)

The solution to the SUSY equations corresponding to the field configuration from (18), (19) can be represented in the following form

3

e = ^ Hs i2 n0 ® n2 (y2 ) ® n2(y3) ® n4(y4). (25)

s=1

/ff ao (r (0) ^3 2 1 ® 2 2

2 ^1 2 1 ® 2 2

2 ^2 ^3 1 <g> 2 2

2 ^2 ^1 i ® 2 2

2 ^2 ^2 1 ® ^3 2

2 ^2 ^2 1 ® ^1 2

2

c

3

Here the operators

ff ff 1o ff2o ff3o

r (0) = r (0)r (0)r (0),

obey

(6)

, ff16 ff26 ff36 i(6)i(6)i(6)

(30)

T(0) =r (6) = ¿1 2, (r (0))2 = (f (6))2 = —12. (31)

We put (f^)) = (ct^ct^ct^, where i = 0, 6 and hence

T(i) = ¿12-

The spinor monomial reads n = n0(x)®X1 ®n1(y1)® n2(y2) ® X2 ® n3(y3) ® n4(y4) ® X3 ® n5(y5) ® ^(^6), where ni = ni(yi) is a 1-component spinor on Mi, i = 1, 2, 3,4, 5 n0 = n0(x) is a 2-component spinor on M^d n6 = n6(y6) is a 2-component spi nor on M6, X^ X2> X3 316 "phantom" spinors.

Here the following relations for modified covariant derivatives take place:

Dm0) = dmo + 4 wa0bomo (^^bo) ^2 ® 1 ® 1 ®

12 ® 1 ® 1 ® 12 ® 1 ® 12),

Dm6) = dm6 + 4wa66}b6m6 (*2 ® 12 ® 1 ® 1 ® 12 ®

1 <g> 1 ®12 ® 1 ® r^f^), (32)

Dmi = dm, for 1, 2, 3, 4, 5 are covariant deriva-

tives corresponding to Mi, i = 0, ^d Dm = dmi, i = 1, 2, 3,4, 5

Here the “factorization” condition

Dmin = ... ® ni-1 ® (D^mlni) ® ni+1 ® ... (33)

is satisfied identically.

M2

read

[s]

p 1i p13 ^p14

— 12 ® O 1 <g) 1 <g) 1 <g) 03 <g) 1 <g) 1 <g) 03 <g) 1 ® 12,

(34)

s = I1

[s]

p 12 p 13 p15

4.3 M2 n M5 n M5

Configuration M2nM5nM5 is defined on manifold

M0 x M1 x M2 x M3 x M4 x M5,

d0 = 2 d1 = 1 d2 = d3 = d4 = d5 = 2

M2

two magnetic ones is given by g = Hi/3H22/3H32/3{ff0 + H-V + H-1ff2 +

+ H2-1H3-1g4 + H^g5}, (39)

F = c1dH-1 A A T3 + c2(*0dH2) A T1 A +

c3(*0dH3) A A T2, (40)

c21 = c22 = c23 = 1 H1 H2 H3 monic functions defined on (M0,g0). The metrics gj (i = 0,1,2,4,5) have Euclidean signatures and the metric g3 has the signature (—, +). The branes sets I1 = {1, 3} I2 = {2, 3, 4} I3 = {3, 4, 5}

We introduce the following set of T-matrices

(fA) = (f^ <g> 1 ® 12 ® I2 ® 12 ® 12,

T(0) <g) 1 <g) f (2) ® f (3) <g> f (4) <g) f (5)

if (0) <g) 1 <g) f ^ ® 12 ® 12 ® 12

r(0) <g> 1 <g> r(2) ® ra‘33) ® 12 ® I2

r(0) <g> 1 <g> r(2) ® r(3) ® raa44) <g> 12

ir(0) ® 1 ® r(2) ® T(3) ® T(4) ® r(41)

where rai) me 2 x 2 r-matrices, ai = 1i, 2^ (i =

0, 2, 3, 4, 5 Mi

operators

^(3) = r(13)r23), r(i) = r1i) r2j),

obey

—12 ® 02 ® 1 ® 1 ® I2 ® 1 ® 1 ® 01 ® 1 ® I2, (35) (r(i)) = 12, (r(3)) = l2, i = 0 2,4, 5.

(42)

(43)

for s = /2. The operator (12) for the M5-brane can be written in the form

f[s] = rio r20 r30 r11 r12 =

— 12 ^12 ® 1 ® 1 <8> ^3 ® 1 ® 1 2 ® 1 <8> I2, (36)

for s = /3.

After a suitable diagonalization of tensor products of Pauli matrices in r [s] we get the following number of unbroken SUSY

Consider n in the form n = n0(x) ® n1 ® n2(y2) ® n3(y3) ® n4(y4) ® n5(y5^, where ni = ni(yi) is a 2Mi i = 0, 2, 3, 4, 5 n1 M1

operator D(i)mi acts on n as

(44)

N = n0n6/32,

(37)

where n0 is the number of parallel spinors on the 3dimensional manifold M^d n6 is the number of chiral parallel spinors on the 2-dimensional manifold M6.

where Dmi is the spinorial covariant derivative corresponding to Mm, i = 0, 2, 3,4,5 mid Di^ = dmi.

M2

[s]

p 1i p13 ^p23

r(0) <g) 1 ® r(2) ® 12 ® r(4) <g> r(5),

s = I1

M5

j = p!op2or 1ip 15p25 =

I2 ® 1 ® r(2) <8> r(3) <8> r(4) <8> I2, s = I2

r [s] = f1o f2° r1i r12 r22 =

I2 ® 1 ® 12 <8> r(3) (g) Ê(4) (g) r(5),

s = I3

The chirality restrictions (16) are satisfied if

r (3)n3 = c(3) nз, c23) = 1

r(j)nj = c(j)nj, c2j) = —1, (45)

j = 0,2,4, 5 and

c(0)c(2)c(4)c(5) = c1, c(2)c(3)c(4) = c2, c(3)c(4)c(5) = c3. (46)

For the field configuration (39) and (40) we ob-

e=

H-1/6H2~1/12H3~1/12n0(x) <g> n1 ® n2(y2) ® n2(y3) ® n4(y4) ® n5(y5)> where n^ i = 0, 2, 3,4,5 are chiral parallel spinors defined on M^ ^D^ml ni = 0), obeying (45) n1

c(0) =

ie4c1c2c3j c(2) = ie5c2c3j c(3) = — e4e5c3j c(4) = ie4j

c(5) = ie4, wher e e4 = ±1 e5 = ±1. Thus, the number of linear independent solutions is

N = 32N =

n0(ie4c1c2c3)n2(ie5 c2c3)n3(—e4e5c3)

£4 = ±1,£5=±1

xn4(ie4)n5(ie5),

where nj (cj ) is the number of chiral parallel spinors on

Mj j = 0, 2, 3, 4, 5

5 Conclusions

As was discussed above, the problem of finding the solutions to SUSY equations is reduced to the search of

Mi

nical) task of finding suitable sets of T-matrices corre-

M

Here we have presented solutions corresponding to various intersecting M-brane configurations of D = 11 supergravity. Using the approach of [3,4] we have found the numbers of preserved supersymmetries for three

M

fiat factor spaces Mi = Rdl we get N =1/8 for any triple configuration in agreement with the classification

of [1].

The presented approach may be of interest from the point of view of possible applications to studies of su-persymmetric solutions defined on product of Ricci-flat manifolds for IIA, IIB supergravities and to supersym-mmetric localized branes.

Acknowledgement

This work was supported by FTsP “Nauchnie i nauchno-pedagogicheskie kadry innovatsionnoy Rossii” for the years 2009-2013. A. G. would like to thank Tomsk State Pedagogical University and the organizers of the Conference "Quantum Field Theory and Gravity (QFTG’12)" for kind hospitality at the Conference.

References

[1] BergshoefF E., de Roo M., Eyras E., Janssen B. and van der Schaar J. P., Class. Quantum Grav., 14, 2757 (1997); [arXiv:hep-th/9612095],

[2] Brecher D. R„ Perry M. J„ 2000 Nucl. Phys. B 556; [arXiv:hep-th/9908018]

[3] Ivashchuk V. D„ 2000, Grav. CosmoL, 6 4; [arXiv:hep-th/0012263],

[4] Ivashchuk V. D„ 2012, IJGMMP 9 8; [arXiv:1107.4089v3 [hep-th]].

[5] Lu H., Pope C. N., Rahmfeld J., 1999, J.Math.Phys. 40 4518-4526; [arXiv:hep-th/9805151]

Received 01.10.2012

А. А. Голубцова, В. Д. Иващук РЕШЕНИЯ ДЛЯ ПЕРЕСЕЧЕНИЯ ТРЕХ БРАН И СУПЕРСИММЕТРИИ

Исследованы композитные М-бранные конфигурации в 11-мерной супергравитации. Суперсимметричные решения описывающие ортогонально пересекающиеся М-бряны определены на. произведении риччи-плоских пространств Mi. Число суперсимметрий зависит от чисел ковариантно-постоянных киральных спиноров на соответствующих фактор-пространствах и знаковых множителей для бран. Представлены точные суперсимметричные решения для трех М-бран на произведении риччи-плоских многообразий со спинорной структурой. Для каждой из конфигураций получено число суперсимметрий и приведены конкретные примеры решений с различными риччи-плоскими фактор-пространствами.

Ключевые слова: М-браны, суперсимметрии, параллельные спиноры, супергравитация.

Голубцова А. А., аспирант.

Российский Университет Дружбы Народов.

Ул. Миклухо-Маклая, б, Москва, Россия, 117198.

Laboratoire de Univers et Théories (LUTh), Observatoire de Paris.

Place Jules Janssen 5, 92190 Meudon, France E-mail: [email protected]

Иващук В. Д., доктор физико-математических наук.

Российский Университет Дружбы Народов.

Ул. Миклухо-Маклая, б, Москва, Россия, 117198.

Центр гравитации и фундаментальной метрологии, VNIIMS.

Озерная, 46, Москва 119361, Россия.

E-mail: [email protected]